fdvposle - Mathbox for Thierry Arnoux

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Theorem fdvposle 32154

Description: Functions with a nonnegative derivative, i.e. monotonously growing functions, preserve ordering. (Contributed by Thierry Arnoux, 20-Dec-2021.)

**Hypotheses**
Ref	Expression
fdvposlt.d	⊢ 𝐸 = (𝐶(,)𝐷)
fdvposlt.a	⊢ (𝜑 → 𝐴 ∈ 𝐸)
fdvposlt.b	⊢ (𝜑 → 𝐵 ∈ 𝐸)
fdvposlt.f	⊢ (𝜑 → 𝐹:𝐸⟶ℝ)
fdvposlt.c	⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ))
fdvposle.le	⊢ (𝜑 → 𝐴 ≤ 𝐵)
fdvposle.1	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≤ ((ℝ D 𝐹)‘𝑥))

**Assertion**
Ref	Expression
fdvposle	⊢ (𝜑 → (𝐹‘𝐴) ≤ (𝐹‘𝐵))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐸 𝑥,𝐹 𝜑,𝑥 Allowed substitution hints: 𝐶(𝑥) 𝐷(𝑥)

**Proof of Theorem fdvposle**
Step	Hyp	Ref	Expression
1		ioossicc 12910	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
3		ioombl 24320	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ dom vol
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
5		fdvposlt.c	. . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ))
6		cncff 23648	. . . . . . . 8 ⊢ ((ℝ D 𝐹) ∈ (𝐸–cn→ℝ) → (ℝ D 𝐹):𝐸⟶ℝ)
7	5, 6	syl 17	. . . . . . 7 ⊢ (𝜑 → (ℝ D 𝐹):𝐸⟶ℝ)
8	7	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ)
9		fdvposlt.d	. . . . . . . 8 ⊢ 𝐸 = (𝐶(,)𝐷)
10		fdvposlt.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐸)
11		fdvposlt.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐸)
12	9, 10, 11	fct2relem 32150	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸)
13	12	sselda 3878	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ∈ 𝐸)
14	8, 13	ffvelrnd 6865	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
15		ioossre 12885	. . . . . . . 8 ⊢ (𝐶(,)𝐷) ⊆ ℝ
16	9, 15	eqsstri 3912	. . . . . . 7 ⊢ 𝐸 ⊆ ℝ
17	16, 10	sseldi 3876	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
18	16, 11	sseldi 3876	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
19		ax-resscn 10675	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
20		ssid 3900	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
21		cncfss 23654	. . . . . . . 8 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ))
22	19, 20, 21	mp2an 692	. . . . . . 7 ⊢ ((𝐴[,]𝐵)–cn→ℝ) ⊆ ((𝐴[,]𝐵)–cn→ℂ)
23	7, 12	feqresmpt 6741	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) = (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)))
24		rescncf 23652	. . . . . . . . 9 ⊢ ((𝐴[,]𝐵) ⊆ 𝐸 → ((ℝ D 𝐹) ∈ (𝐸–cn→ℝ) → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ)))
25	12, 5, 24	sylc 65	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D 𝐹) ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
26	23, 25	eqeltrrd 2835	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℝ))
27	22, 26	sseldi 3876	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
28		cniccibl 24596	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿¹)
29	17, 18, 27, 28	syl3anc 1372	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿¹)
30	2, 4, 14, 29	iblss 24560	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑥)) ∈ 𝐿¹)
31	7	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (ℝ D 𝐹):𝐸⟶ℝ)
32	2	sselda 3878	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ (𝐴[,]𝐵))
33	32, 13	syldan 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ 𝐸)
34	31, 33	ffvelrnd 6865	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
35		fdvposle.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≤ ((ℝ D 𝐹)‘𝑥))
36	30, 34, 35	itgge0 24566	. . 3 ⊢ (𝜑 → 0 ≤ ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥)
37		fdvposle.le	. . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
38		fdvposlt.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐸⟶ℝ)
39		fss 6522	. . . . 5 ⊢ ((𝐹:𝐸⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐸⟶ℂ)
40	38, 19, 39	sylancl 589	. . . 4 ⊢ (𝜑 → 𝐹:𝐸⟶ℂ)
41		cncfss 23654	. . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ))
42	19, 20, 41	mp2an 692	. . . . 5 ⊢ (𝐸–cn→ℝ) ⊆ (𝐸–cn→ℂ)
43	42, 5	sseldi 3876	. . . 4 ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ))
44	9, 10, 11, 37, 40, 43	ftc2re 32151	. . 3 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑥) d𝑥 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
45	36, 44	breqtrd 5057	. 2 ⊢ (𝜑 → 0 ≤ ((𝐹‘𝐵) − (𝐹‘𝐴)))
46	38, 11	ffvelrnd 6865	. . 3 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
47	38, 10	ffvelrnd 6865	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
48	46, 47	subge0d 11311	. 2 ⊢ (𝜑 → (0 ≤ ((𝐹‘𝐵) − (𝐹‘𝐴)) ↔ (𝐹‘𝐴) ≤ (𝐹‘𝐵)))
49	45, 48	mpbid 235	1 ⊢ (𝜑 → (𝐹‘𝐴) ≤ (𝐹‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ⊆ wss 3844 class class class wbr 5031 ↦ cmpt 5111 dom cdm 5526 ↾ cres 5528 ⟶wf 6336 ‘cfv 6340 (class class class)co 7173 ℂcc 10616 ℝcr 10617 0cc0 10618 ≤ cle 10757 − cmin 10951 (,)cioo 12824 [,]cicc 12827 –cn→ccncf 23631 volcvol 24218 𝐿¹cibl 24372 ∫citg 24373 D cdv 24618

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-inf2 9180 ax-cc 9938 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-pre-sup 10696 ax-addf 10697 ax-mulf 10698

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-symdif 4134 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-iin 4885 df-disj 4997 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-se 5485 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-isom 6349 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-of 7428 df-ofr 7429 df-om 7603 df-1st 7717 df-2nd 7718 df-supp 7860 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-2o 8135 df-oadd 8138 df-omul 8139 df-er 8323 df-map 8442 df-pm 8443 df-ixp 8511 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-fsupp 8910 df-fi 8951 df-sup 8982 df-inf 8983 df-oi 9050 df-dju 9406 df-card 9444 df-acn 9447 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-9 11789 df-n0 11980 df-z 12066 df-dec 12183 df-uz 12328 df-q 12434 df-rp 12476 df-xneg 12593 df-xadd 12594 df-xmul 12595 df-ioo 12828 df-ioc 12829 df-ico 12830 df-icc 12831 df-fz 12985 df-fzo 13128 df-fl 13256 df-mod 13332 df-seq 13464 df-exp 13525 df-hash 13786 df-cj 14551 df-re 14552 df-im 14553 df-sqrt 14687 df-abs 14688 df-limsup 14921 df-clim 14938 df-rlim 14939 df-sum 15139 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-starv 16686 df-sca 16687 df-vsca 16688 df-ip 16689 df-tset 16690 df-ple 16691 df-ds 16693 df-unif 16694 df-hom 16695 df-cco 16696 df-rest 16802 df-topn 16803 df-0g 16821 df-gsum 16822 df-topgen 16823 df-pt 16824 df-prds 16827 df-xrs 16881 df-qtop 16886 df-imas 16887 df-xps 16889 df-mre 16963 df-mrc 16964 df-acs 16966 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-submnd 18076 df-mulg 18346 df-cntz 18568 df-cmn 19029 df-psmet 20212 df-xmet 20213 df-met 20214 df-bl 20215 df-mopn 20216 df-fbas 20217 df-fg 20218 df-cnfld 20221 df-top 21648 df-topon 21665 df-topsp 21687 df-bases 21700 df-cld 21773 df-ntr 21774 df-cls 21775 df-nei 21852 df-lp 21890 df-perf 21891 df-cn 21981 df-cnp 21982 df-haus 22069 df-cmp 22141 df-tx 22316 df-hmeo 22509 df-fil 22600 df-fm 22692 df-flim 22693 df-flf 22694 df-xms 23076 df-ms 23077 df-tms 23078 df-cncf 23633 df-ovol 24219 df-vol 24220 df-mbf 24374 df-itg1 24375 df-itg2 24376 df-ibl 24377 df-itg 24378 df-0p 24425 df-limc 24621 df-dv 24622

This theorem is referenced by: fdvnegge 32155

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